Nonlinear wave dynamics in self-consistent water channels

Abstract

We study long-wave dynamics in a self-consistent water channel of variable cross-section, taking into account the effects of weak nonlinearity and dispersion. The self-consistency of the water channel is considered within the linear shallow water theory, which implies that the channel depth and width are interrelated, so the wave propagates in such a channel without inner reflection from the bottom even if the water depth changes significantly. In the case of small-amplitude weakly dispersive waves, the reflection from the bottom is also small, which allows the use of a unidirectional approximation. A modified equation for Riemann waves is derived for the nondispersive case. The wave-breaking criterion (gradient catastrophe) for self-consistent channels is defined. If both weak nonlinearity and dispersion are accounted for, the variable-coefficient Korteweg–de Vries (KdV) equation for waves in self-consistent channels is derived. Note that this is the first time that a KdV equation has been derived for waves in strongly inhomogeneous media. Soliton transformation in a channel with an abrupt change in depth is also studied.

Non-Reflective Magnetohydrodynamic Waves in an Inhomogeneous Plasma

“…which is known for an isothermal atmosphere with the effective height H/6.7 [13], where H is the equivalent height of the magnetic field variation.…”

“…At the same time, their use gave an answer to the question about the existence of traveling linear waves in a one-dimensional inhomogeneous medium and made it possible to find the characteristics of the inhomogeneous medium itself. A variety of physical examples was considered for waves on the surface of shallow water [12,13], long internal waves at the interface of two liquids [14,15], acoustic waves in the atmospheres of the Earth and the Sun [16,17], waves in solar magnetic tubes [18][19][20][21], elastic waves [22], and the waves generated when passing through gradient barriers [23]. In the case of spatial problems, similar traveling waves were obtained for waveguide systems with a uniform distribution of the field across the waveguide, where essentially the problem reduces to one-dimensional [13,20,22,24].…”

“…A variety of physical examples was considered for waves on the surface of shallow water [12,13], long internal waves at the interface of two liquids [14,15], acoustic waves in the atmospheres of the Earth and the Sun [16,17], waves in solar magnetic tubes [18][19][20][21], elastic waves [22], and the waves generated when passing through gradient barriers [23]. In the case of spatial problems, similar traveling waves were obtained for waveguide systems with a uniform distribution of the field across the waveguide, where essentially the problem reduces to one-dimensional [13,20,22,24]. It was shown that the number of non-reflective waveguide configurations can be large enough, so that the existence of traveling waves is not uncommon.…”

“…It was shown that the number of non-reflective waveguide configurations can be large enough, so that the existence of traveling waves is not uncommon. There are only a few examples for the nonlinear problems [13,24].…”

Asymptotic, Convergent, and Exact Truncating Series Solutions of the Linear Shallow Water Equations for Channels with Power Law Geometry

“…The cases where ray theory is exact can be conceived as cases for which the variable coefficient equations may be transformed to a standard wave equation with constant coefficients. A slightly more general approach transforms the shallow water equations to a constant coefficient Klein-Gordon equation [15,9,24]. However, this and a few other exact solutions of the shallow water equations with non-constant depth, such as the oscillations in parabolic basins [32], are less closely related to the present work.…”